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Grothendieck Topologies and Étale Cohomology

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Grothendieck Topologies and Étale Cohomology Grothendieck topologies and étale cohomology Pieter Belmans My gratitude goes to prof. Bruno Kahn for all the help in writing these notes. And I would like to thank Mauro Porta, Alexandre Puttick, Mathieu Rambaud for spotting some errors in a previous version of this text. Contents 1 Grothendieck topologies2 1.1 Motivation....................................2 1.2 Definitions....................................2 1.3 First examples..................................3 1.4 Sheaves on Grothendieck topologies....................4 1.5 More examples..................................5 1.6 Comparison of cohomology..........................6 2 Étale cohomology9 2.1 Motivation....................................9 2.2 The étale topology...............................9 2.3 Galois cohomology............................... 10 2.4 Weil cohomologies............................... 11 2.5 Constructible sheaves.............................. 13 2.6 Glueing...................................... 14 3 Results in étale cohomology 16 3.1 Proper base change............................... 16 3.2 Higher direct images with proper support................. 17 3.3 Smooth base change.............................. 19 3.4 Purity....................................... 21 3.5 Poincaré duality................................. 22 3.6 Grothendieck’s six operations......................... 23 3.7 A few words on the literature......................... 23 1 1 Grothendieck topologies 1.1 Motivation Around 1950 the Zariski topology was introduced in algebraic geometry, in order to have a topology that is appropriate for the objects (i.e. varieties) that were studied, unlike the Euclidean topology. In the late 1950s this was generalized to schemes. In 1949 Weil had proposed conjectures, now named after him, relating prop- erties of algebraic varieties over finite fields to the topological properties of their counterparts over C. These conjectures have been discussed before in class, and the case of curves has been proven. At some point it was realized that the existence of a “Weil cohomology theory”, mimicking the properties of algebraic topology, can solve the Weil conjectures. This observation is probably due to Serre, who attributes it himself to Weil. But in algebraic topology one often uses constant sheaves. Unfortunately the Zariski topology is not adapted to these sheaves as the following proposition shows. Proposition 1. Let X be an irreducible topological space. Let F be a constant sheaf on X . Then i (1) H (X , F) = 0 for i 1. ≥ Proof. Every nonempty open set U of X is connected, so F is a flabby sheaf. Therefore all higher cohomology vanishes. This applies in particular to (irreducible) schemes with the Zariski topology. So we conclude that to define a topology on a scheme which gives a meaningful cohomology theory for constant sheaves we need to find something different. When discussing the motivation for the étale topology in section 2.1 another bad property of the Zariski topology is given. But remark that the Zariski topology is already the good topology to calculate the so-called coherent cohomology of quasicoherent sheaves: these cohomology groups will be isomorphic for all the subcanonical topologies discussed in section 1.5. 1.2 Definitions The notion of a Grothendieck topology is a very natural one (albeit maybe in hindsight). To realize this we consider the basic definitions of sheaf theory. Recall that a presheaf F on a topological space X is an assignment (2) U F(U) 7! of sets (or (abelian) groups, rings, modules, . ) to every open set U of the space, together with restriction morphisms resV,U for V U. In the functorial language this is nothing but a functor on the category of open⊆ sets of X , where morphisms correspond to inclusions. A sheaf is a separated presheaf satisfying the glueing property, i.e. it is com- pletely determined by its local data. The separatedness implies that for every open cover U S U and sections f , g F U such that for all i we have f g we = i I i ( ) Ui = Ui 2 2 j Sj have f = g globally. And the glueing condition says that if we are given U = i I Ui 2 2 and sections f F U such that f f then there exists a section f i ( i) i Ui Uj = j Ui Uj \ \ on U restricting2 to fi on each Ui. j j Again, this can be interpreted in purely categorical terms: intersections are actually fiber products, and the glueing property can be taken as the exactness of the equaliser Y Y (3) F(U) F(Ui) ⇒ F(Ui U Uj) ! i I i,j I × 2 2 if the category of values of F has products. So there is no need to restrict oneself to topological spaces for sheaf theory: as long as the category shares some properties with the category of open sets of a topological space (or rather: gives information similar to open covers!) one can generalize without any problem. Definition 2. Let C be a category. A Grothendieck topology on C consists of sets of morphisms Ui U which are called covers for each object U such that f ! g 1. if V U is an isomorphism the singleton V U is a cover; ! f ! g 2. if Ui U is a cover, and V U is a morphism, then all the fibered productsf !Ui g U V exist, and set of! induced projections Ui U V V is again a cover; × f × ! g 3. if Ui U is a cover, and for each i we have a cover Vi,j Ui then the set off compositions! g Vi,j U is again a cover. f ! g f ! g When a category C is equipped with a Grothendieck topology we call it a site. These axioms don’t describe a topology using open sets, but in terms of covers. In the classical notion of a topology one needs to check that the given description of its open sets satisfies some properties with respect to intersections and unions. In the case of a Grothendieck topology on the other hand one considers preservation under base change and composition. Some examples in algebraic geometry come to mind: open immersions, étale morphisms, smooth morphisms, . Remark that the terminology (which is the one found in [ALB73, definition 1.1.1]) doesn’t completely agree with the terminology in [SGA31; SGA41]. Just like different bases for a topological space can induce the same topology, this definition defines what is called a pretopology. Different pretopologies can induce the same topology, and hence the associated sheaf theory on the sites is the same. 1.3 First examples Example 3. Take a topological space X . The category of its open sets, i.e. the objects are open sets and arrows are inclusions, is equipped with a Grothendieck topology. To each open subset U of X we associate the collection of open covers of U. Fibered products of inclusions are intersections. So the objects act (or in this case: are) “open sets”, but the most important thing are the morphisms. These describe “how” the “open set” is “contained” in the space. This is a so called “small” example. We can also equip the whole category of topological spaces (or schemes) with a Grothendieck topology. To do so we first introduce an important notion. Definition 4. Let Ui U be a cover in a site in which set-theoretic unions make sense (topologicalf spaces,! g schemes, . ). It is jointly surjective if the set-theoretic union of the images equals U. 3 Now we can change our focus to algebraic geometry. Example 5. The small Zariski site of a scheme X is the category XZar which is the full subcategory of Sch=X of objects U X that are open immersions equipped with a Grothendieck topology by defining! a cover Ui X to be a jointly surjective set of open embeddings. f ! g And we also have a bigger version. Example 6. The big Zariski site (Sch=X )Zar of a scheme X is the category Sch=X equipped with a Grothendieck topology by defining a cover Ui U to be a jointly surjective set of open embeddings. f ! g There are also the étale versions. Example 7. The small étale site of a scheme X is the category Xét which is the full subcategory of Sch=X of objects U X that are étale and locally of finite presentation equipped with a Grothendieck! topology by defining a cover Ui X to be a jointly surjective set of étale morphisms. If U X and V Xf are! twog objects of Xét every arrow U V in Xét is necessarily étale.! ! ! Remark 8. If one takes X = Spec k the small étale site yields already interesting properties. In case of the Zariski topology there is only one open set, the space itself. But if k is not separably closed we can consider a separable extension and this yields an étale morphism, which is immediately jointly surjective, so it yields a cover. We can also consider products of separable extensions. Hence studying the small étale site of a point is equivalent to studying (products of) separable extensions and their tensor products, which is exactly what Galois cohomology is about. Example 9. The big étale site (Sch=X )ét of a scheme X is the category Sch=X equipped with a Grothendieck topology by defining a cover Ui U to be a jointly surjective set of étale morphisms that are locally of finitef presentation.! g One might feel uncomfortable with the existence of two different sites, the small one containing the “opens” of X while the big one contains strictly more information. But at least for the Zariski and étale topology there is no difference in the cohomology groups (after we’ve introduced sheaf theory and sheaf cohomology). Remark also that for some topologies there is no small site (for instance the cdh topology, introduced in section 1.5).
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